I am learning about conformal mapping, and I found it hard to answer question from the kind "determine if there exist mobius transformation between $A$ and $B$" for example: $$A=\left\{z \in \mathbb{C} \mid \operatorname{Im}(z)>0\right\}$$ $$B=\left\{z \in D_{1}(0) \mid \operatorname{Re}(z)>0\right\}$$
why there is not exist such transformation between those $2$ domains ? How should I approach this kind of questions?
How should I aproach this kind of questions? Thanks in advance